Degree (angle)
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- This article describes "degree" as a unit of angle. For alternative meanings, see degree.
A degree (in full, a degree of arc, arc degree, or arcdegree), usually symbolized °, is a measurement of plane angle, representing 1⁄360 of a full rotation. When that angle is with respect to a reference meridian, it indicates a location along a great circle of a sphere (such as Earth, Mars, or the celestial sphere).[1]
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[edit] History
The number 360 as the number of 'degrees' (or minimal/practical sub-arcs) in a circle, and hence the unit of a degree as a sub-arc of 1⁄360 of the circle, was probably adopted because it approximates the number of days in a year. Ancient astronomers noticed that the stars in the sky, which circle the celestial pole every day, seem to advance in that circle by approximately one-360th of a circle, i.e. one degree, each day. Primitive calendars, such as the Persian Calendar used 360 days for a year. Its application to measuring angles in geometry can possibly be traced to Thales who popularized geometry among the Greeks and lived in Anatolia (modern western Turkey) among people who had dealings with Egypt and Babylon.
This division is used in mathematics, but also in geography and in astronomy to measure the celestial sphere and equator (both in terms of latitude and longitude).
[edit] India
The division of the circle into 360 degrees dates back to ancient India, as found in the Rig Veda:
- Twelve spokes, one wheel, navels three.
- Who can comprehend this?
- On it are placed together
- three hundred and sixty like pegs.
- They shake not in the least.
- (Dirghatama, Rig Veda 1.164.48)
[edit] Further justification
The number 360 is useful since it is readily divisible: 360 has 24 divisors (including 1 and 360), including every number from 1 to 10 except 7. For the number of degrees in a circle to be divisible by every number from 1 to 10, there would need to be 2520 degrees in a circle, which is a much less convenient number.
- Divisors of 360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
For many practical purposes, a degree is a small enough angle that whole degrees provide sufficient precision. When this is not the case, as in astronomy or for latitudes and longitudes on the Earth, degree measurements may be written with decimal places, but the traditional sexagesimal unit subdivision is commonly seen. One degree is divided into 60 minutes (of arc), and one minute into 60 seconds (of arc). These units, also called the arcminute and arcsecond, are respectively represented as a single and double prime, or if necessary by a single and double closing quotation mark: for example, 40.1875° = 40° 11' 15". If still more accuracy is required, decimal divisions of the second are normally used, rather than thirds of 1⁄60 second, fourths of 1⁄60 of a third, and so on. These (rarely used) subdivisions were noted by writing the Roman numeral for the number of sixtieths in superscript: 1I for a "prime" (minute of arc), 1II for a second, 1III for a third, 1IV for a fourth, etc. Hence the modern symbols for the minute and second of arc.
[edit] Alternative units
In mathematics, angles in degrees are rarely used, as the convenient divisibility of the number 360 is not so important. For various reasons, mathematicians typically prefer to use the radian. In this system the angles 180° and π radians are equal, or equivalently, the degree is a mathematical constant ° = π⁄180. This means, that in a complete circle (360°) there are 2π radians. The circumference of a circle is 2πr, where r is the radius.
With the invention of the metric system, based on powers of ten, there was an attempt to define a "decimal degree" (grad or gon), so that the number of decimal degrees in a right angle would be 100 gon, and there would be 400 gon in a circle. Although this idea did not gain much momentum, most scientific calculators support it.
An angular mil which is most used in military applications has at least three specific variants.
In computer games which depict a 3 dimensional virtual world, the need for very fast computations resulted in the adoption of a binary, 256 degree system. In this system, a right angle is 64 degrees, angles can be represented in a single byte, and all trigonometric functions are implemented as small lookup tables. There is currently no known name for this unit.
[edit] enneagram
The enneagram expresses the circle as equal to 9 integers. The structure of the enneagram is based partly on the primary triangle of the circle at 0/360 degrees, 120 degrees and 240 degrees of the circle. In terms of integers, these points of the circle correspond to the numbers 0/9, 3 and 6. The rest of the Enneagram structure consists of connections between the other 6 integers of the 9-based number system - determined by the fraction 1/7 = 0.142857 (repeating).
[edit] See also
[edit] References
- ^ Beckmann P. (1976) A History of Pi, St. Martin's Griffin. ISBN 0-312-38185-9
[edit] External links
- Degrees as an angle measure, with interactive animation
- Degree at MathWorld
